Almost periodic solution of a discrete competitive system with delays and feedback controls

Definition 1

(see [11]). A sequence x : Z → R^k is called an almost periodic sequence if the ε-translation set of x

is a relatively dense set in Z for all ε > 0; that is, for any given ε > 0, there exists an integer l(ε) > 0 such that each discrete interval of length l(ε) contains an integer τ = τ(ε) ∈ E{ε, x} such that

Definition 2

(see [11]). Let f : Z × D → R^k, where D is an open set in C = {φ : [−τ, 0]_Z → R^k}. f(n, φ) is said to be almost periodic in n uniformly for φ ∈ D, if for any ε > 0 and any compact set S ⊂ D, there exists a positive integer l = l(ε, S), such that any interval of length l = l(ε, S) contains an integer τ, for which

Definition 3

(see [12]). The hull of f, denoted by H(f), is defined by

for some sequence τ_k, where S is any compact set in D.

Lemma 4

(see [13]). {x(n)} is an almost periodic sequence if and only if for any integer sequence {ki′} , there exists a subsequence {k_i} ⊂ {ki′} such that x(n + k_i) converges uniformly on n ∈ Z as i → ∞. Furthermore, the limit sequence is also an almost periodic sequence.

Lemma 5

(see [14]). Assume that r(n) > 0, {x(n)} satisfies x(n) > 0, and

for n ∈ [n₁, +∞), where a is a positive constant. Then

Lemma 6

(see [14]). Assume that r(n) > 0, {x(n)} satisfies x(n) > 0, and

for n ∈ [n₁, +∞), lim supn→+∞ x(n) ≤ x^*, and x(n₁) > 0, where a and x^* are positive constants such that ax^* > 1. Then

Lemma 7

(see [15]). Assume that A > 0 and x(0) > 0, and further suppose that

then, for any integer k ≤ n,

Specifically, if A < 1 and B(n) is bounded above with respect to M, then

Lemma 8

(see [15]). Assume that A > 0 and x(0) > 0, and further suppose that

then, for any integer k ≤ n,

Specifically, if A < 1 and B(n) is bounded below with respect to m, then

Lemma 9

(see [11]). Suppose that there exists a Lyapunov function V(n, φ, ψ) satisfying the following conditions:

1. a(∣φ(0) − ψ(0)∣) ≤ V(n, φ, ψ) ≤ b(∥φ−ψ∥), where a, b ∈ P with P = {a : [0, +∞) → [0, +∞), a(0) = 0 and a(u) is continuous, increasing in u}.

2. V(n, φ₁, ψ₁ − V(n, φ₂, ψ₂) ≤ L(∥φ₁ − φ₂∥ + ∥ψ₁ − ψ₂∥), where L > 0 is a constant.

3. △ V(n, φ, ψ) ≤ −α V(n, φ, ψ), where 0 < α < 1 is a constant.

Remark 10

(see [16]). Condition (iii) of Lemma 9 can be replaced by:

1. △ V(n, φ, ψ) ≤ −β V(n, φ, ψ), where β ∈ {c : [0, +∞) → [0, +∞), c is continuous, c(0) = 0 and c(s) > 0 for s > 0}.

Theorem 11

Assume that

hold, then the system (2) is permanent. i.e., there exist positive constants m_i, M_i, h_i and H_i, such that for any positive solution (x₁(k), ⋯,x_n(k), u₁(k), ⋯, u_n(k)) of system (2), one has

Proof

From system (2),

By Lemma 5, we know that

We can choose a sufficiently small ε such that for large enough K₁ > 0, we have

For k > K₁, we have

As a direction corollary of Lemma 7, one has

Setting ε → 0,

For above ε, there exists an integer K₂ > K₁ such that

From (23), (27) and system (2),

where △iε=ril−∑j=1,j≠inbiju(Mj+ε)−∑j=1,j≠indiju(Mj+ε)2−eiu(Hi+ε)>0. Noting the fact that exp(x − 1) > x, for x > 0, we have

Then by Lemma 6, one has

Setting ε → 0,

There exists a positive integer K₃ > K₂+τ such that

From (32) and system (2), we have

By Lemma 8, one has

Setting ε → 0, it follows that

Denoting

Theorem 12

Assume that the condition (19) holds, then Ω ≠ Φ.

Proof

Since the coefficients are almost periodic sequences, there exists an integer valued sequence {t_p} with t_p → ∞ as p → ∞ such that

We can choose a sufficiently small ε. From Theorem 11, there exists a positive integer N₀ such that

Denoting x_ip(k) = x_i(k+t_p), u_ip(k) = u_i(k+t_p) for k > N₀ − t_p and p = 1, 2, ⋯. For any positive integer q, it is easy to see that there exists a sequence {x_ip(k): p ≥ q} such that the sequence {x_ip(k)} has a subsequence, also denoted by {x_ip(k)}, converging on any finite interval A of Z⁺ as p → ∞.

In fact, for any finite subset A = {l₁, l₂, ⋯, l_m} ⊆ Z⁺, where m is a finite number, t_p+l_j > N₀ (j = 1, 2, ⋯, m), when p is large enough. Therefore m_i−ε ≤ x_i(k+t_p) ≤ M_i+ε (i = 1, 2, ⋯, n); that is, x_i(k+t_p) are uniformly bounded when p is sufficiently large. Next, for l₁ ∈ A, we choose a subsequence {tp(1)} of {t_p} such that xi(k+tp(1)) uniformly converge on Z⁺ for p sufficiently large. Similar to the arguments of l₁, for l₂ ∈ A, one can select a subsequence {tp(2)} of {tp(1)} such that xi(k+tp(2)) uniformly converge on Z⁺ for p sufficiently large. Repeating above-mentioned process, for l_m ∈ A, one obtains a subsequence {tp(m)} of {tp(m−1)} such that xi(k+tp(m)) uniformly converge on Z⁺ for p sufficiently large.

Based on the above, one selects the sequence {tp(m)} which is a subsequence of {t_p}, still denoted by {t_p}, then one gets x_i(k+t_p) → xi∗ uniformly in k ∈ A as k → ∞. So the conclusion holds truly due to the arbitrariness of A. Thus we have a sequence {y_i(k)} such that for k ∈ Z⁺,

which, together with (36), yields that

It follows from (36), (38) and (39) that

It is easy to see that (y₁(k), ⋯,y_n(k), v₁(k), ⋯, v_n(k)) is a solution of system (2) and m_i−ε ≤ y_i(k) ≤ M_i+ε, h_i−ε ≤ v_i(k) ≤ H_i+ε for k ∈ [−τ, 0]_Z.

Theorem 13

Assume that the condition (19) and

hold, where i, j = 1, 2, ⋯, n. Then there exists a unique uniformly asymptotically stable almost periodic solution (x₁(k), ⋯, x_n(k), u₁(k), ⋯, u_n(k)) of system (2) which is bounded by Ω for all k ∈ N⁺.

Proof

Let ω_i(k) = ln x_i(k), i = 1, 2, ⋯, n, then system (2) can be rewritten as

From Theorem 12, there exists a solution (ω₁(k), ⋯, ω_n(k), u₁(k), ⋯, u_n(k)) of system (42) such that

which implies that ∣ω_i(k)∣ ≤ B_i = max{∣ ln m_i∣, ∣ ln M_i∣} and ∣u_i(k)∣ ≤ C_i = max{h_i, H_i}, i = 1, 2, ⋯, n. For k ∈ Z⁺ and s ∈ [−τ, 0]_Z, assign

are two solutions of system (42) defined on D,

Defining

then ∥W_k(s)∥ ≤ E = ∑i=1n [B_i+C_i].

Consider the product system of (42)

Construct the Lyapunov functional V(k),

Apparently,

Denoting

On the other side,

Thus condition (i) in Lemma 9 is satisfied if we take a(x) = x and b(x) = ρ x, where a, b ∈ C(R⁺, R⁺).

By using (3.2) in Ref. [16], for any W_k, Z_k, W̄_k, Z̄_k ∈ D, we have

so, condition (ii) in Lemma 9 is also satisfied.

By the mean value theorem, it derives that

where θ_i(k) lie between ω_i(k) and z_i(k), and η_i(k − τ_i),ξ_i(k − τ_i) all lie between ω_i(k − τ_i) and z_i(k − τ_i), respectively. Then

Calculating the △ V(k) along with the solution of system (47), we have